Convergence in Lp space for the homogenization problems of elliptic and parabolic equations in the plane

We study the convergence rate of an asymptotic expansion for the elliptic and parabolic operators with rapidly oscillating coefficients. First we propose homogenized expansions which are convolution forms of Green function and given force term of elliptic equation. Then, using local Lp-theory, the growth rate of the perturbation of Green function is found. From the representation of elliptic solution by Green function, we estimate the convergence rate in Lp space of the homogenized expansions to the exact solution. Finally, we consider L2(0,T : H1(Ω)) or L∞(Ω × (0,T )) convergence rate of the first order approximation for parabolic homogenization problems.  2003 Elsevier Inc. All rights reserved.